Upper d-Ahlfors measure, Hausdoff $N-1$ dimensional measure, usual surface measure

Question

I'd like to know the relation between Upper d-Ahlfors measure , Hausdorff $N-1$ dimensional measure and usual surface measure.

(1) Let $\Omega$ be a smooth bounded domain. Then, on $\partial \Omega,$ do two measures coincide with the usual surface measure?

(2) Are there any relations about 3 measures above?

I would be grateful for any comment about it.

Answer 1

Well, if the open set, then they coincide on the bounduary since it is rectifiable (to be fair its reduced bounduary is rectifiable).

I think in general knowing that your surface is the bonduary of an open set does not give you any further information and I think it may be possible they differ.

A little remark: on rectifiable sets all reasonable surface measures coincide.